Solve 9/r^2-3r+21/r^2+8r-1/r^2+5r-24

$$\frac { 9 } { r ^ { 2 } - 3 r } + \frac { 21 } { r ^ { 2 } + 8 r } - \frac { 1 } { r ^ { 2 } + 5 r - 24 }$$

Evaluate

$$\frac{29r+9}{r\left(r-3\right)\left(r+8\right)}$$

Short Solution Steps

Factor r^{2}-3r. Factor r^{2}+8r.

$$\frac{9}{r\left(r-3\right)}+\frac{21}{r\left(r+8\right)}-\frac{1}{r^{2}+5r-24}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r\left(r-3\right) and r\left(r+8\right) is r\left(r-3\right)\left(r+8\right). Multiply \frac{9}{r\left(r-3\right)} times \frac{r+8}{r+8}. Multiply \frac{21}{r\left(r+8\right)} times \frac{r-3}{r-3}.

$$\frac{9\left(r+8\right)}{r\left(r-3\right)\left(r+8\right)}+\frac{21\left(r-3\right)}{r\left(r-3\right)\left(r+8\right)}-\frac{1}{r^{2}+5r-24}$$

Since \frac{9\left(r+8\right)}{r\left(r-3\right)\left(r+8\right)} and \frac{21\left(r-3\right)}{r\left(r-3\right)\left(r+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{9\left(r+8\right)+21\left(r-3\right)}{r\left(r-3\right)\left(r+8\right)}-\frac{1}{r^{2}+5r-24}$$

Do the multiplications in 9\left(r+8\right)+21\left(r-3\right).

$$\frac{9r+72+21r-63}{r\left(r-3\right)\left(r+8\right)}-\frac{1}{r^{2}+5r-24}$$

Combine like terms in 9r+72+21r-63.

$$\frac{30r+9}{r\left(r-3\right)\left(r+8\right)}-\frac{1}{r^{2}+5r-24}$$

Factor r^{2}+5r-24.

$$\frac{30r+9}{r\left(r-3\right)\left(r+8\right)}-\frac{1}{\left(r-3\right)\left(r+8\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r\left(r-3\right)\left(r+8\right) and \left(r-3\right)\left(r+8\right) is r\left(r-3\right)\left(r+8\right). Multiply \frac{1}{\left(r-3\right)\left(r+8\right)} times \frac{r}{r}.

$$\frac{30r+9}{r\left(r-3\right)\left(r+8\right)}-\frac{r}{r\left(r-3\right)\left(r+8\right)}$$

Since \frac{30r+9}{r\left(r-3\right)\left(r+8\right)} and \frac{r}{r\left(r-3\right)\left(r+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{30r+9-r}{r\left(r-3\right)\left(r+8\right)}$$

Combine like terms in 30r+9-r.

$$\frac{29r+9}{r\left(r-3\right)\left(r+8\right)}$$

Expand r\left(r-3\right)\left(r+8\right).

$$\frac{29r+9}{r^{3}+5r^{2}-24r}$$

Differentiate w.r.t. r

$$\frac{2\left(108-45r-86r^{2}-29r^{3}\right)}{\left(r\left(r-3\right)\left(r+8\right)\right)^{2}}$$